I. Field of the Invention
The present invention relates to a method for removing phase errors in digitized Magnetic Resonance Imaging (MRI) data resulting from quadrature demodulation of an MRI echo signal from an observed object.
II. Background Information
A Two Dimensional Fourier Transform MRI system such as that discussed by A. Kumar, et al. in NMR Fourier Zeugmatography, J. Magn. Reson. 18:69-83, 1975, subjects an object, such as a human body, to a series of magnetic fields and radio frequency (RF) pulses. The RF pulses resonate selected atoms in the object as a consequence of the characteristics of the atoms, the magnetic fields, and the RF pulses. The resonant atoms precess in the magnetic fields and radiate a detectable RF echo signal containing information about the location of the radiating atoms. As is well known to those skilled in the art, the echo, when quadrature demodulated by a reference signal, is the two dimensional Fourier transform (2D FT), designated F(X,Y), of the signal intensity per unit area, designated f(x,y), of the selected atoms in a slice through the object. The function f(x,y) represents the spatial distribution of the resonant atoms and their relaxation times in the object, and is used to construct a visual image of a slice through the object. Because of unknown and time varying differences between the phase of the echo signal from the object and the quadrature demodulation reference signal, the demodulation process introduces a time varying phase error signal into the MRI data that distorts the final visual image.
In the 2D FT technique, the object, shown oriented in a space coordinate system in FIG. 1, is subjected to a static magnetic field, usually parallel to the z axis. In addition, the object is also subjected to gradient magnetic fields. In a typical gradient field, the direction of the field is parallel to the static field, but the magnitude of the field at any point in space is proportional to either the x, y, or z coordinate. By proper selection of the magnitudes and durations of the magnetic gradient fields, data points represented by the function G(X,Y) can be collected along specific paths of the Fourier Space plane defined by: ##EQU1## where g.sub.x (t) is the "read" magnetic field gradient with a gradient direction parallel to the x-axis, g.sub.y (t) is the "encode" magnetic field gradient with a gradient parallel to the y-axis, and .gamma. is the magnetogyric ratio of the atoms under observation. These gradients are typically measured in "field per unit length," for example T/cm, and the equations assume that a time origin t=0 has been suitably defined.
To produce the measured data function, G(X,Y), the RF echo signal from the object must be quadrature demodulated. Because of instabilities in the frequency of the quadrature demodulation reference signal oscillator, a phase shift error that varies with time in an unknown manner is introduced into the measured data. Because of this phase error, G(X,Y) is related to the desired data by the relationship: EQU G(X,Y)=e.sup.i.PHI.(m) .multidot.F(X,Y),
where e.sup.i.PHI.(m) is the phase error introduced by the demodulation process, m is the number of the path parallel to the X-axis in Fourier space and is related to Y by the equation: EQU Y=m.multidot..DELTA.Y
where .DELTA.Y is the Fourier space distance between successive paths, and F(X,Y) is the desired data function corrected for phase errors, which may be transformed into f(x,y) and the final visual image.
FIG. 2 shows a series of 2D FT stepwise paths in Fourier space. The time needed to scan along one path is very short, being on the order of tens of milliseconds, as compared to the time needed to make successive scans on each step, which takes on the order of a few seconds per step. A scan through one slice of an object comprising many steps on the Fourier plane takes time on the order of several minutes, during which time the quadrature demodulation reference oscillator may vary in phase and frequency by a significant, but unknown amount. Because f(x,y) is real, F(X,Y) is real at the origin of Fourier space and the phase of the complex number F(O,O) is zero. Thus, in the absence of the phase errors, e.sup.i.PHI.(m), the phase of G(O,O) would also be zero. Any phase term measured in G(O,O) must, therefore, be the phase error resulting from the demodulation process, and that error can be calculated at the origin. However, the phase error varies in an unknown manner with time, and the value calculated at the origin of Fourier space may not apply at large distances from the origin, representing relatively long time periods.
Phase correction is straightforward in projection reconstruction (PR) scans but is a problem in 2D FT scans. In the PR scan, every echo collected is a central section of the transform and is therefore known to be real at the origin. Because of this, the phase error can be calculated and removed. For the echoes in a 2D FT scan, this central section property is not true, and the same correction is not applicable. As noted above, during the time period of an entire scan, the phase error calculated at the origin does not necessarily apply to data at remote locations from the origin, since the demodulation oscillator may vary with time by a significant, but unknown amount during the long time period needed to take successive scan echoes.
Accordingly, there exists a need for a technique to correct quadrature demodulation phase errors in 2D FT signals over the entire expanse of a Fourier space plane, that takes into consideration reference signal oscillator instabilities over the relatively long time periods needed to measure multiple scans through an entire slice of an object.